Trigonometry: From Goniometry to Triangles

This page presents a comprehensive overview of trigonometric concepts and their application to triangles. The content is organized to provide a clear understanding of various theorems and formulas essential in trigonometry.

Vocabulary: Goniometry refers to the measurement of angles and the study of angular functions.

The page begins by establishing conventions for labeling triangle elements:

• Vertices are denoted by uppercase letters
• Sides are represented by lowercase letters
• Angles are indicated using Greek letters

Definition: In trigonometry, a right-angled triangle is a triangle containing one 90-degree angle.

The document then proceeds to outline several key theorems:

1. First Theorem (for right-angled triangles): This theorem likely refers to the basic trigonometric ratios in right-angled triangles, though specific formulas are not provided in the image.

2. Second Theorem: The following formulas are presented:

   • b = c tan β
   • c = a cos β
   • b = a cos α

Example: In a right-angled triangle, if the hypotenuse (c) is 10 units and angle β is 30°, then side b can be calculated as b = 10 * tan(30°) ≈ 5.77 units.

3. Chord Theorem: The formula a = 2r sin α is provided, where 'r' likely represents the radius of a circle and 'a' the length of a chord.

4. Law of Sines: The formula (a / sin α) = (b / sin β) = (c / sin γ) is presented, which is applicable to all triangles.

Highlight: The Law of Sines is a fundamental theorem in trigonometry, allowing for the solution of triangles when certain side lengths and angles are known.

5. Consequences of the two theorems:

   • The formula (side₁ * side₂) / 2 = r² * sin(included angle) is provided.
   • For an inscribed angle in a semicircle: If ABC is inscribed in a semicircle with AC as the diameter, then angle ABC = 90°.

6. Carnot's Theorem (Law of Cosines): Three equivalent formulas are presented:

   • b² = a² + c² - 2ac cos β
   • a² = b² + c² - 2bc cos α
   • c² = a² + b² - 2ab cos γ

Vocabulary: The Law of Cosines is also known as the Teorema di Carnot in Italian, named after the French mathematician Lazare Carnot.

The page concludes with a note about 'r' representing the radius of the circumscribed circle of the triangle.

This comprehensive summary covers the essential teoremi sui triangoli rettangoli and teoremi trigonometria triangoli qualsiasi, providing a solid foundation for understanding and applying trigonometric concepts to various triangle problems.